# Two Red Lines in a Rectangle

Imagine a rectangle with sides $$\overline{\text{AB}}$$, $$\overline{\text{BC}}$$, $$\overline{\text{CD}}$$, and $$\overline{\text{DA}}$$. Side $$\overline{\text{AB}}$$ has a length of $$4$$ while $$\overline{\text{BC}}$$ has a length of $$6$$.

Put point $\text{E}$ on $\overline{\text{AD}}$. Draw a line from $\text{E}$ to $\text{B}$ and draw another line from $\text{E}$ to $\text{C}$.

If $\angle\text{BEC}$ has a measurement of $65°$, what is the length of line segments $\overline{\text{EB}}$ and $\overline{\text{EC}}$?

Rectangle $\text{ABCD}$

How can I solve for the lengths of the red lines? A step-by-step solution will be very much appreciated.

Here's the link of an image of described shape: http://imgur.com/gallery/x6kGdAq

Note by Kaizen Cyrus
5 months, 2 weeks ago

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Here's a hint: let $\overline{DE}$ be $x$. Then, $\overline{AE} = 6-x$. We can use the Pythagorean Theorem to express $\overline{EB}$ and $\overline{EC}$ in terms of $x$. Then, you can use Cosine Law on $\triangle EBC$ and solve for $x$. Plug the result back into $\overline{EB}$ and $\overline{EC}$, and the problem is solved!

- 5 months, 2 weeks ago

$\overline{AE}$ is equal to $6-x$? But $\overline{AD}$ is $6$ and you said let it be $x$.

- 5 months, 2 weeks ago

Whoops, my mistake. I meant let $\overline{DE}$ be $x$. Everything else stays the same.

- 5 months, 1 week ago